Spectral methods for the spin-2 equation near the cylinder at spatial infinity

Macedo, Rodrigo Panosso; Kroon, Juan A. Valiente

doi:10.1088/1361-6382/aac116

Cited by 8 publications

(6 citation statements)

References 47 publications

(101 reference statements)

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“…In this section, we use Friedrich's cylinder at spatial infinity construction in conjunction with Theorem 3.8 to obtain the existence of solutions to a class of semilinear wave equations near spatial infinity in Minkowski spacetime. For other applications of the cylinder at spatial infinity construction to linear wave and spin-2 equations on Minkowski spacetime, see [7,17,19,27,44].…”

Section: Wave Equations On Minkowski Spacetime Near Spatial Infinitymentioning

confidence: 99%

The Fuchsian approach to global existence for hyperbolic equations

Beyer¹,

Oliynyk²,

Olvera-Santamaría³

2019

Preprint

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We analyze the Cauchy problem for symmetric hyperbolic equations with a time singularity of Fuchsian type and establish a global existence theory along with decay estimates for evolutions towards the singular time under a small initial data assumption. We then apply this theory to semilinear wave equations near spatial infinity on Minkowski and Schwarzschild spacetimes, and to the relativistic Euler equations with Gowdy symmetry on Kasner spacetimes.Constants of this type will always be non-negative, non-decreasing, continuous functions of their arguments.Given four vector bundles V , W , Y and Z that sit over Σ, and mapsFor situations, where we want to bound f (t, w, v) by g(t, v) up to an undetermined constant of proportionality, we defineif there exists a R ∈ (0, R) and a map f ∈ C 0 [T 0 , 0), C ∞ (B R (W ) × B R (V ), L(Y, Z))such that

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Section: Wave Equations On Minkowski Spacetime Near Spatial Infinitymentioning

confidence: 99%

The Fuchsian approach to global existence for hyperbolic equations

Beyer¹,

Oliynyk²,

Olvera-Santamaría³

2019

Preprint

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“…In this paper, we continue investigations of the conformally invariant wave equation, which was studied on a Minkowski background in [11] and on a Schwarzschild spacetime in [12,13]. As examples for other related problems discussed in the literature, we also refer to numerical solutions of the spin-2 equation on Minkowski [4,5,9,23] and analytical investigations of the Maxwell equations on Schwarzschild [34,35].…”

Section: Introductionmentioning

confidence: 98%

Fully pseudospectral solution of the conformally invariant wave equation on a Kerr background

Hennig¹,

Macedo²

2021

Class. Quantum Grav.

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We study axisymmetric solution to the conformally invariant wave equation on a Kerr background by means of numerical and analytical methods. Our main focus is on the behaviour of the solutions near spacelike infinity, which is appropriately represented as a cylinder. Earlier studies of the wave equation on a Schwarzschild background have revealed important details about the regularity of the corresponding solutions. It was found that, on the cylinder, the solutions generically develop logarithmic singularities at infinitely many orders. Moreover, these singularities also ‘spread’ to future null infinity. However, by imposing certain regularity conditions on the initial data, the lowest-order singularities can be removed. Here we are interested in a generalisation of these results to a rotating black hole background and study the influence of the rotation rate on the properties of the solutions. To this aim, we first construct a conformal compactification of the Kerr solution which yields a suitable representation of the cylinder at spatial infinity. Besides analytical investigations on the cylinder, we numerically solve the wave equation with a fully pseudospectral method, which allows us to obtain highly accurate numerical solutions. This is crucial for a detailed analysis of the regularity of the solutions. In the Schwarzschild case, the numerical problem could effectively be reduced to solving (1 + 1)-dimensional equations. Here we present a code that can perform the full 2 + 1 evolution as required for axisymmetric waves on a Kerr background.

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“…In this paper, we continue investigations of the conformally invariant wave equation, which was studied on a Minkowski background in [9] and on a Schwarzschild spacetime in [10,11]. As examples for other related problems discussed in the literature, we also refer to numerical solutions of the spin-2 equation on Minkowski [4,5,7,20] and analytical investigations of the Maxwell equations on Schwarzschild [30,31].…”

Section: Introductionmentioning

confidence: 92%

Fully pseudospectral solution of the conformally invariant wave equation on a Kerr background

Hennig,

Macedo

2020

Preprint

View full text Add to dashboard Cite

We study axisymmetric solution to the conformally invariant wave equation on a Kerr background by means of numerical and analytical methods. Our main focus is on the behaviour of the solutions near spacelike infinity, which is appropriately represented as a cylinder. Earlier studies of the wave equation on a Schwarzschild background have revealed important details about the regularity of the corresponding solutions. It was found that, on the cylinder, the solutions generically develop logarithmic singularities at infinitely many orders. Moreover, these singularities also 'spread' to future null infinity. However, by imposing certain regularity conditions on the initial data, the lowest-order singularities can be removed. Here we are interested in a generalisation of these results to a rotating black hole background and study the influence of the rotation rate on the properties of the solutions. To this aim, we first construct a conformal compactification of the Kerr solution which yields a suitable representation of the cylinder at spatial infinity. Besides analytical investigations on the cylinder, we numerically solve the wave equation with a fully pseudospectral method, which allows us to obtain highly accurate numerical solutions. This is crucial for a detailed analysis of the regularity of the solutions. In the Schwarzschild case, the numerical problem could effectively be reduced to solving (1 + 1)-dimensional equations. Here we present a code that can perform the full 2 + 1 evolution as required for axisymmetric waves on a Kerr background.

show abstract

Spectral methods for the spin-2 equation near the cylinder at spatial infinity

Cited by 8 publications

References 47 publications

The Fuchsian approach to global existence for hyperbolic equations

The Fuchsian approach to global existence for hyperbolic equations

Fully pseudospectral solution of the conformally invariant wave equation on a Kerr background

Fully pseudospectral solution of the conformally invariant wave equation on a Kerr background

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